Question

Let the following interest rate 5.80% be considered a nominal interest rate.

In your responses below, provide two decimal places (with proper rounding) and do NOT include the percent (%) sign. For example, to provide the response "two and one quarter percent," you would enter 2.25 (and not 2.25% or 0.0225).

Given the nominal rate above, what is the effective interest rate when the compounding is:

semiannual?  

quarterly?  

monthly?  

Answer 1

Solution:

Effective interest rate is calculated using the following formula:

EAR = ( 1 + ( r/n) )n – 1

Where

r = Nominal Interest rate ; n = Number of compounding periods ;

Calculation of Effective Interest Rate when compounding is semiannual :

As per the information given in the question we have

r = 5.8 % = 0.58 ; n = 12 / 6 = 2 periods

Thus applying the above values in the formula we have

EAR = ( 1 + ( 0.58 / 2 ) )2 – 1

= ( 1 + ( 0.029) )2 – 1

= ( 1.029 )2 – 1

= 1.058841 – 1

= 0.058841

= 5.8841 %

= 5.88 % ( when rounded off to two decimal places )

Note : ( 1.029 )2 is calculated using the excel formula

=POWER(Number,Power) = POWER(1.029,2) = 1.058841

Calculation of Effective Interest Rate when compounding is quarterly :

As per the information given in the question we have

r = 5.8 % = 0.58 ; n = 12 / 3 = 4 periods

Thus applying the above values in the formula we have

EAR = ( 1 + ( 0.58 / 4 ) )4 – 1

= ( 1 + ( 0.014500 )4 – 1

= ( 1.014500 )4 – 1

= 1.059274 – 1 = 0.059274

= 5.9274 %

= 5.93 % ( when rounded off to two decimal places )

Note : ( 1.014500 )4 is calculated using the excel formula

=POWER(Number,Power) = POWER(1.014500,4) = 1.059274

Calculation of Effective Interest Rate when compounding is monthly :

As per the information given in the question we have

r = 5.8 % = 0.58 ; n = 12 / 1 = 12 periods

Thus applying the above values in the formula we have

EAR = ( 1 + ( 0.58 / 12 ) )12 – 1

= ( 1 + ( 0.004833 ) )12 – 1

= ( 1.004833 )12 – 1

= 1.059567 – 1 = 0.059567

= 5.9567 %

= 5.96 % ( when rounded off to two decimal places )

Note : ( 1.004833 )12 is calculated using the excel formula

=POWER(Number,Power) = POWER(1.004833,12) = 1.059567

Thus Given the nominal rate of 5.80 %, the effective interest rate when the compounding is:

Semiannual = 5.88

Quarterly = 5.93

Monthly = 5.96